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f(x) = -4x^3 + ax^2 + 9x - 18, where a is a constant - Edexcel - A-Level Maths Pure - Question 6 - 2014 - Paper 1
Question 6
f(x) = -4x^3 + ax^2 + 9x - 18, where a is a constant. Given that (x - 2) is a factor of f(x), a) find the value of a, b) factorise f(x) completely, c) find the r... show full transcript
Worked Solution & Example Answer:f(x) = -4x^3 + ax^2 + 9x - 18, where a is a constant - Edexcel - A-Level Maths Pure - Question 6 - 2014 - Paper 1
Step 1
a) find the value of a
Answer
To determine the value of a, we substitute x = 2 into f(x) since (x - 2) is a factor. Therefore,
f(2) = -4(2)^3 + a(2)^2 + 9(2) - 18.
Calculating this gives:
f(2) = -32 + 4a + 18 - 18
f(2) = 4a - 32.
Since (x - 2) is a factor, f(2) = 0:
0 = 4a - 32,
Solving for a:
4a = 32
a = 8.
Step 2
b) factorise f(x) completely
Answer
Substituting the value of a into f(x) yields:
f(x) = -4x^3 + 8x^2 + 9x - 18.
To factorise completely, we will first factor by grouping:
= -4(x^3 - 2x^2 - rac{9}{4}x + rac{9}{2}).
Next, we can use synthetic division or polynomial division with the factor (x - 2):
This results in:
f(x) = -(x - 2)(4x^2 + 1).
Since 4x^2 + 1 cannot be factored further over the reals, the complete factorisation is:
f(x) = -(x - 2)(4x^2 + 1).
Step 3
c) find the remainder when f(x) is divided by (2x - 1)
Answer
To find the remainder when f(x) is divided by (2x - 1), we can use the Remainder Theorem. First, we need to find the value of x for which 2x - 1 = 0. Solving this gives:
2x = 1
x = rac{1}{2}.
Next, we evaluate f( rac{1}{2}):
figg( rac{1}{2}igg) = -4igg( rac{1}{2}igg)^3 + 8igg( rac{1}{2}igg)^2 + 9igg( rac{1}{2}igg) - 18.
Calculating this:
= -4 imes rac{1}{8} + 8 imes rac{1}{4} + rac{9}{2} - 18
= - rac{1}{2} + 2 + rac{9}{2} - 18
= rac{10}{2} - 18 = 5 - 18 = -13.
Thus, the remainder when f(x) is divided by (2x - 1) is -13.